IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v169y2016i2d10.1007_s10957-016-0881-6.html
   My bibliography  Save this article

Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

Author

Listed:
  • Dario Bauso

    (The University of Sheffield
    Università di Palermo)

  • Quanyan Zhu

    (Polytechnic School of Engineering New York University)

  • Tamer Başar

    (University of Illinois at Urbana-Champaign)

Abstract

Mixed integer optimal compensation deals with optimization problems with integer- and real-valued control variables to compensate disturbances in dynamic systems. The mixed integer nature of controls could lead to intractability in problems of large dimensions. To address this challenge, we introduce a decomposition method which turns the original n-dimensional optimization problem into n independent scalar problems of lot sizing form. Each of these problems can be viewed as a two-player zero-sum game, which introduces some element of conservatism. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon, a step that mirrors a standard procedure in mixed integer programming. We apply the decomposition method to a mean-field coupled multi-agent system problem, where each agent seeks to compensate a combination of an exogenous signal and the local state average. We discuss a large population mean-field type of approximation and extend our study to opinion dynamics in social networks as a special case of interest.

Suggested Citation

  • Dario Bauso & Quanyan Zhu & Tamer Başar, 2016. "Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 606-630, May.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:2:d:10.1007_s10957-016-0881-6
    DOI: 10.1007/s10957-016-0881-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-016-0881-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-016-0881-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hamidou Tembine & Quanyan Zhu & Tamer Basar, 2011. "Risk-sensitive mean field stochastic differential games," Post-Print hal-00643547, HAL.
    2. POCHET, Yves & WOLSEY, Laurence A., 1993. "Lot-sizing with constant batches: formulation and valid inequalities," LIDAM Reprints CORE 1066, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Sebastian Sager & Mathieu Claeys & Frédéric Messine, 2015. "Efficient upper and lower bounds for global mixed-integer optimal control," Journal of Global Optimization, Springer, vol. 61(4), pages 721-743, April.
    4. Andrew J. Clark & Herbert Scarf, 2004. "Optimal Policies for a Multi-Echelon Inventory Problem," Management Science, INFORMS, vol. 50(12_supple), pages 1782-1790, December.
    5. Yves Pochet & Laurence A. Wolsey, 1993. "Lot-Sizing with Constant Batches: Formulation and Valid Inequalities," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 767-785, November.
    6. Daron Acemoğlu & Giacomo Como & Fabio Fagnani & Asuman Ozdaglar, 2013. "Opinion Fluctuations and Disagreement in Social Networks," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 1-27, February.
    7. repec:dau:papers:123456789/3001 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Stefanny Ramirez & Dario Bauso, 2021. "Dynamic Coordination Games with Activation Costs," Dynamic Games and Applications, Springer, vol. 11(3), pages 580-596, September.
    2. Stefanny Ramirez & Dario Bauso, 2023. "Dynamic Games with Strategic Complements and Large Number of Players," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 1-21, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Laurence A. Wolsey, 2002. "Solving Multi-Item Lot-Sizing Problems with an MIP Solver Using Classification and Reformulation," Management Science, INFORMS, vol. 48(12), pages 1587-1602, December.
    2. Gaetan Belvaux & Laurence A. Wolsey, 2001. "Modelling Practical Lot-Sizing Problems as Mixed-Integer Programs," Management Science, INFORMS, vol. 47(7), pages 993-1007, July.
    3. Jans, Raf & Degraeve, Zeger, 2007. "Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1855-1875, March.
    4. BELVAUX, Gaetan & WOLSEY, Laurence A., 2000. "Modelling practical lot-sizing problems as mixed integer programs," LIDAM Discussion Papers CORE 2000009, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Stefanny Ramirez & Dario Bauso, 2021. "Dynamic Coordination Games with Activation Costs," Dynamic Games and Applications, Springer, vol. 11(3), pages 580-596, September.
    6. Archibald, Thomas W. & Bokkers, Menno B. & Dekker, Rommert & Vliet, Andre van, 1999. "Minimising bins in transmission systems," European Journal of Operational Research, Elsevier, vol. 115(2), pages 380-391, June.
    7. Gaetan Belvaux & Laurence A. Wolsey, 2000. "bc --- prod: A Specialized Branch-and-Cut System for Lot-Sizing Problems," Management Science, INFORMS, vol. 46(5), pages 724-738, May.
    8. Miguel Constantino, 1998. "Lower Bounds in Lot-Sizing Models: A Polyhedral Study," Mathematics of Operations Research, INFORMS, vol. 23(1), pages 101-118, February.
    9. Hark-Chin Hwang, 2010. "Economic Lot-Sizing for Integrated Production and Transportation," Operations Research, INFORMS, vol. 58(2), pages 428-444, April.
    10. STEPHAN, Rüdiger, 2010. "An extension of disjunctive programming and its impact for compact tree formulations," LIDAM Discussion Papers CORE 2010045, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    11. Chung-Lun Li & Qingying Li, 2016. "Polynomial-Time Solvability of Dynamic Lot Size Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(03), pages 1-20, June.
    12. Hark-Chin Hwang, 2009. "Inventory Replenishment and Inbound Shipment Scheduling Under a Minimum Replenishment Policy," Transportation Science, INFORMS, vol. 43(2), pages 244-264, May.
    13. Akbalik, Ayse & Rapine, Christophe, 2013. "The single item uncapacitated lot-sizing problem with time-dependent batch sizes: NP-hard and polynomial cases," European Journal of Operational Research, Elsevier, vol. 229(2), pages 353-363.
    14. Alper Atamtürk & Dorit S. Hochbaum, 2001. "Capacity Acquisition, Subcontracting, and Lot Sizing," Management Science, INFORMS, vol. 47(8), pages 1081-1100, August.
    15. van Hoesel, C.P.M. & Wagelmans, A.P.M., 1997. "Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems," Econometric Institute Research Papers EI 9735/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    16. Meisel, Frank & Kirschstein, Thomas & Bierwirth, Christian, 2013. "Integrated production and intermodal transportation planning in large scale production–distribution-networks," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 60(C), pages 62-78.
    17. Hwang, Hark-Chin & Kang, Jangha, 2016. "Two-phase algorithm for the lot-sizing problem with backlogging for stepwise transportation cost without speculative motives," Omega, Elsevier, vol. 59(PB), pages 238-250.
    18. François Vanderbeck, 1998. "Lot-Sizing with Start-Up Times," Management Science, INFORMS, vol. 44(10), pages 1409-1425, October.
    19. Mathieu Van Vyve, 2005. "The Continuous Mixing Polyhedron," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 441-452, May.
    20. Akbalik, A. & Pochet, Y., 2009. "Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs," European Journal of Operational Research, Elsevier, vol. 198(2), pages 412-434, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:169:y:2016:i:2:d:10.1007_s10957-016-0881-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.